Extending structures and classifying complements for left-symmetric algebras

الأصل · EN

Let A be a left-symmetric (resp. Novikov) algebra, E be a vector space containing A as a subspace and V be a complement of A in E.The extending structures problem which asks for the classification of all left-symmetric (resp. Novikov) algebra structures on E such that A is a subalgebra of E is studied. In this paper, the definition of the unified product of left-symmetric (resp. Novikov) algebras is introduced. It is shown that there exists a left-symmetric (resp. Novikov) algebra structure on E such that A is a subalgebra of E if and only if E is isomorphic to a unified product of A and V. Two cohomological type objects Hₐ²(V,A) and H²(V,A) are constructed to give a theoretical answer to the extending structures problem. Furthermore, given an extension A⊂ E of left-symmetric (resp. Novikov) algebras, another cohomological type object is constructed to classify all complements of A in E. Several special examples are provided in details.